↑

Properties

Label 1-416-416.179-r1-0-0
Degree $1$
Conductor $416$
Sign $-0.309 + 0.950i$
Analytic cond. $44.7054$
Root an. cond. $44.7054$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.707 + 0.707i)5-s + (−0.866 + 0.5i)7-s + (0.866 − 0.5i)9-s + (−0.258 − 0.965i)11-s + (0.5 − 0.866i)15-s + (0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.707 − 0.707i)21-s + (0.866 + 0.5i)23-s − i·25-s + (−0.707 + 0.707i)27-s + (0.965 − 0.258i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.707 + 0.707i)5-s + (−0.866 + 0.5i)7-s + (0.866 − 0.5i)9-s + (−0.258 − 0.965i)11-s + (0.5 − 0.866i)15-s + (0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.707 − 0.707i)21-s + (0.866 + 0.5i)23-s − i·25-s + (−0.707 + 0.707i)27-s + (0.965 − 0.258i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.309 + 0.950i$
Analytic conductor: \(44.7054\)
Root analytic conductor: \(44.7054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 416,\ (1:\ ),\ -0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4164959082 + 0.5734179782i\)
\(L(\frac12)\) \(\approx\) \(0.4164959082 + 0.5734179782i\)
\(L(1)\) \(\approx\) \(0.6154077192 + 0.1551251011i\)
\(L(1)\) \(\approx\) \(0.6154077192 + 0.1551251011i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (-0.258 + 0.965i)T \)
67 \( 1 + (0.965 - 0.258i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.4032072418806971771273599491, −23.07999952766765154405128723932, −22.470471473041332904688048076407, −21.06471311856354468360709346012, −20.360570658138875256569596326, −19.34906318227408124028094352268, −18.613481429004464190868151985902, −17.54154167869657724011660945957, −16.67004304115569290265118161764, −16.13338170175313253525881961828, −15.30417893185965147426456455238, −13.8552193617259528151173259184, −12.768847815296463798521684149742, −12.34114360380275296157694540534, −11.455874361323431642777547891196, −10.263834408793215215941109486756, −9.6107633857751672591374702760, −8.12667171771144069597207395293, −7.23585804211524145399759821425, −6.43649979039052088200398299557, −5.100419450854384073751117325051, −4.46540573595025029369378180121, −3.15379167314611057479594080539, −1.35642949377341069455077411104, −0.33765952256659670459264018740, 0.80973088091795200584116350978, 2.84550305618319718910571364825, 3.644561013686399805204493536133, 4.92557145189395564314117201827, 6.06661633818481088243092608735, 6.65732176035618486895120253587, 7.82778592523669379900632676158, 9.06222266667443945936908849040, 10.1716948135078768024163830789, 10.949627037932108222614015017635, 11.7202848723352764342466022762, 12.571103985583973477178440434869, 13.57032734341902729370734261166, 14.94649848919885885770978111366, 15.68457093660013854402830127672, 16.250721712180371214137291522062, 17.29829267990314509141038035177, 18.26597876804518546873080764966, 19.11479674668088743840534716106, 19.601047969388708196525039238085, 21.31838843516456110958691693664, 21.71477721303025769379990878213, 22.68198011294786950516211317685, 23.25860109775911844808719461780, 24.028589047084076548333675582068

Graph of the $Z$-function along the critical line